Question

The area of a rhombus if its vertices are (3,4),(5,-1),(-1,0),(3,1) taken in order is ………………..square units​

Answer 1

Step-by-step explanation:

Given :-

The vertices of a rhombus are (3,4),(5,-1),(-1,0),(3,1)

To find :-

Find the area of the rhombus ?

Solution :-

Given that

The vertices of a rhombus are (3,4),(5,-1),(-1,0),(3,1)

Consider a rhombus PQRS

Let P = (3,4)

Let Q = (5,-1)

Let R = (-1,0)

Let S = (3,1)

We know that

Area of a rhombus = (1/2)d1×d2 sq.units

Where, d1 and d2 are the diagonals

We have ,PR and QS are the diagonals

Length of PR:-

Let (x1, y1) = (3,4) => x1 = 3 and y1 = 4

Let (x2, y2) = (-1,0) => x2 = -1 and y2 = 0

We know that

The distance between two points (x1, y1) and (x2, y2) is √[(x2-x1)²+(y2-y1)²] units

=> Distance between P and R

=> PR = √[(-1-3)²+(0-4)²]

=> PR = √[(-4)²+(-4)²]

=> PR = √(16+16)

=> PR = √32

=> PR = √(2×16)

=> PR = 4√2 units

Length of QS :-

Let (x1, y1) = (5,-1) => x1 = 5 and y1 = -1

Let (x2, y2) = (3,1) => x2 = 3 and y2 = 1

We know that

The distance between two points (x1, y1) and (x2, y2) is √[(x2-x1)²+(y2-y1)²] units

=> Distance between Q and S

=> QS = √[(3-5)²+(1-(-1))²]

=> QS = √[(-2)²+(2)²]

=> QS = √(4+4)

=> QS = √8

=> QS = √(2×4)

=> QS = 2√2 units

Now,

The area of the given rhombus PQRS

=> (1/2)× PR×QS

=> (1/2)×(4√2)×(2√2) sq.units

=> (1/2)×(4×2×√2×√2) sq.units

=> (1/2)×8×2 sq units

=> 16/2 sq.units

=> 8 sq.units

Therefore, Area = 8 sq.units

Answer:-

Area of the given rhombus is 8 sq.units

Used formulae:-

→ Area of a rhombus = (1/2)d1×d2 sq.units

Where, d1 and d2 are the diagonals

Distance formula:-

→ The distance between two points (x1, y1) and (x2, y2) is √[(x2-x1)²+(y2-y1)²] units